Velocity of Swiftly Moving Electrified Particles. 591 



appears that it is possible from the calculations to draw some 

 conclusions of importance for the comparison of the theory 

 with the measurements. 



Consider a /3 particle passing through a sheet of matter, 

 and let us for a moment assume that no collision occurs for 

 which \ is smaller than a certain value t. Let the value 

 for jo determined from (14) by putting X = t be p T . If t is 

 not small compared with unity the probability distribution 

 of the loss of energy will with considerable approximation 

 be given by (8), if in the expression for P the integral is 

 performed from p—pr instead of from p = 0. According to 

 the above p T will be great compared with a, and we get 

 instead of the expression (9) for P 



Pr=~ -^ (10) 



Introducing this in (11) we find for a sheet of aluminium 

 0*01 gr. per cm. 2 for u approximately «r = 250r. If r is not 

 small compared with unity, we therefore see that we obtain 

 a probability distribution of the loss of energy which is of 

 the same character as that for a rays. The mean value for 

 the loss of energy for the collisions in question is simply 

 obtained from the formula (5) in the former section by re- 

 placing a by p r . This gives 



^.-^i, <; ). . . . (16) 



In the applications the logarithmic term in this formula 

 will be large and A r T will depend very little upon the exact 

 value of t. Thus for an aluminium sheet A r T will vary 

 only 4 per cent., if t varies from 1 to 2. 



Let us now consider the probability distribution of the 

 loss of energy due to the collisions for which p is smaller 

 than ;> r . Since p r is large compared with a, it follows from 

 (14) that the average number of these collisions is very 

 nearly equal to t. If now r is a small number, e. g. t=1, it 

 is evident that the probability distribution of the loss of 

 energy due to the collisions will be of a type quite different 

 from that considered above. In the first place, there is a 

 certain probability that there will be no loss of energy at all; 

 from (6) we get that this probability is equal to e _r . Next, 

 if Qr is the value given by (1) if we put p=p T , no loss of 

 energy greater than zero and smaller than Q r is possible. 

 At Qr the probability curve suddenly rises and falls off for 

 increasing values of Q approximately as Q~ 2 . For the 



